Year 10 Higher · Simultaneous equations
Solve a pair of equations by drawing their graphs
Learn how to solve simultaneous equations by graphs, elimination and substitution, including pairs where a straight line meets a curve.
Definition
Simultaneous equations are two equations about the same two unknowns that must both be true at the same time. Solving them means finding the values that satisfy both at once, which is the point, or points, where their graphs cross. Two straight lines can cross at most once, so a linear pair has at most one solution. When one equation describes a curve such as y = x², a straight line can cross it twice, so the pair can have two solutions.
Why Do I Have to Learn This?
Simultaneous equations solve two mysteries at once, like the price of two different snacks. They pop up whenever two things depend on each other.
Key Rules:
- 1. The solutions of a pair of simultaneous equations are the crossing points of their graphs: two straight lines cross at most once, but a line and a curve can cross twice.
Remember:
- •Every method finds where the graphs cross: line up the coefficients, eliminate or substitute, and check both values in both original equations.
Don’t Forget...
- •Adding the equations when the matched terms have the same sign, or subtracting when the signs are opposite. +6y and -6y cancel by adding, but +6y and +6y cancel by subtracting; the wrong choice doubles the term to 12y instead of removing it, leaving two unknowns.
- •Multiplying the left-hand side of an equation but forgetting the right-hand side. An equation only stays true when every term on both sides is multiplied: 3x + 2y = 19 times 3 is 9x + 6y = 57, not 9x + 6y = 19.
- •Stopping after one x value when the quadratic from a line and a curve has two roots. A line can cross a curve twice, so each root is a separate crossing point, and each x must be paired with its own y value, never with the y from the other point.
Worked Example:
Solve the simultaneous equations 3x + 2y = 19 and 2x - 3y = 4.
No coefficients match, so multiply the first equation by 3 and the second by 2: 9x + 6y = 57 and 4x - 6y = 8.
The y terms are +6y and -6y, opposite signs, so add the equations: 13x = 65, giving x = 5.
Substitute x = 5 into 3x + 2y = 19: 15 + 2y = 19, so 2y = 4 and y = 2.
Check in the other original equation: 2 × 5 - 3 × 2 = 10 - 6 = 4, which matches.
Answer: x = 5, y = 2
Why It Works:
Multiplying an equation through by a number, adding equals to equals, and replacing an unknown by an expression it equals all turn true statements into true statements, so any solution of the original pair must survive every step. Elimination is valid because both equations hold at the same pair of values, so their sum or difference must hold there too. Substituting a line into a curve produces a quadratic that is true exactly where both graphs agree, so its roots are precisely the crossing points.
Another Example:
The curve y = x² and the straight line y = x + 6 cross at two points. Find the coordinates of both points.
Both equations give y, so set the two expressions equal: x² = x + 6.
Rearrange to zero and factorise: x² - x - 6 = 0, so (x - 3)(x + 2) = 0.
The roots are x = 3 and x = -2; substitute each into y = x + 6 to get y = 9 and y = 4.
Check both points on the curve: 3² = 9 and (-2)² = 4, so each point lies on both graphs.
Answer: (3, 9) and (-2, 4)
Animated Example:
Free Solve a pair of equations by drawing their graphs Worksheet
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Before this: Equations, inequalities and formulae (Year 9) · Next: Equations and formulae (Year 11 Higher)
Let’s Practise the Concept
Step 1 of 3
Order the data
Tap the values from smallest to largest.
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